Critical depth in open-channel flow (rectangular channel): which expression gives the critical depth h_c in terms of discharge per unit width q?

Introduction / Context:
Critical depth is the depth at which the specific energy is minimum for a given discharge in an open channel. For a rectangular channel, it provides a convenient criterion for transitions between subcritical (tranquil) and supercritical (torrential) flow and is widely used in the design of flumes and spillways.

Given Data / Assumptions:

• Rectangular channel of unit width; discharge per unit width q = Q / b.
• Gravitational acceleration g constant.
• Frictional losses neglected in the local specific energy analysis.

Concept / Approach:

Specific energy E for a rectangular channel is E = h + q^2/(2 * g * h^2). The critical depth h_c occurs when dE/dh = 0, giving the condition Fr = 1 (q^2 = g * h_c^3), from which h_c is obtained.

Step-by-Step Solution:

Verification / Alternative check (if short method exists):

Use Froude number for rectangular section: Fr = v / sqrt(g * h) with v = q / h. Setting Fr = 1 → (q / h) = sqrt(g * h) → q^2 = g * h^3, identical result.

Why Other Options Are Wrong:

They invert or misapply powers of q and g, or confuse dimensional consistency (depth must scale as (q^2/g)^(1/3)).

Common Pitfalls (misconceptions, mistakes):

Forgetting that q is discharge per unit width, not total discharge; misusing Froude number definition.

Final Answer:

h_c = ( q^2 / g )^(1/3)